In electronics a NOT gate is more commonly called an inverter. The circle on the symbol is called a bubble, and is used in logic diagrams to indicate a logic negation between the external logic state and the internal logic state (1 to 0 or vice versa).
A weather watch, on the other hand, typically refers to an alert issued to indicate that conditions are favorable for the development of dangerous weather patterns, although the dangerous weather conditions themselves are not currently present.
Winter weather watches and warnings. A winter storm watch is issued when winter storm conditions are possible within the next 12 to 48 hours but the timing intensity or occurrence may still be uncertain. A winter storm warning is issued when heavy snow is occurring or will develop in the next 18 hours. The heavy snow may be accompanied by wind greater than 15 mph and blowing snow. A blizzard watch is issued when blizzard conditions are possible in the next 12 to 24 hours.